Strategy-proof division with single-peaked preferences and initial endowments

We consider the problem of (re)allocating the total endowment of an infinitely divisible commodity among agents with single-peaked preferences and initial endowments. We propose an extension of the so-called uniform rule and show that it is the unique rule satisfying strategy-proofness, Pareto optimality, and an equal-treatment condition. This last condition is implied by the combination of anonymity and translation invariance, which fact is used to obtain a second characterization. The resulting rule turns out to be peaks-only and individually rational: the allocation assigned by the rule depends only on the peaks of the preferences, and no agent is worse off than at his initial endowment.